Question

For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]

Answer 1

We have,

\[\frac{dy}{dx} = y \tan x\]

\[ \Rightarrow \frac{1}{y}dy = \tan x dx\]

Integrating both sides, we get

\[\int\frac{1}{y}dy = \int\tan x dx\]

\[ \Rightarrow \log y = \log \left| \sec x \right| + C . . . . . . . \left( 1 \right)\]

Now,

When `x = 0, y = 1`

\[ \therefore \log 1 = \log 1 + C\]

\[ \Rightarrow C = 0\]

Putting the value of `C` in (1), we get

\[\log y = \log \left| \sec x \right|\]

\[ \Rightarrow y = \sec x\]